Superstring Scattering from O-planes

0.21 MB·

by  Mohammad R. Garousi

Superstring Scattering
from O-planes
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0
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2 Mohammad R. Garousi

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a
Department of Physics, Ferdowsi university, P.O. Box 1436, Mashhad, Iran
J

6 and

Institute for Studies in Theoretical Physics and Mathematics IPM
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v P.O. Box 19395-5531, Tehran, Iran
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7
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6 Abstract
0
/ We write the vertex operators of massless NS-NS and RR states of TypeII super-
h
t string theory in the presence of Orientifold p-planes. They include the usual vertex
-
p operators of Type II theory and their images. We then calculate the two-point func-
e tions of these vertex operators at the projective plane PR level. We show that the
h 2
: resultcan bewritten in the Veneziano-type formulae, with the same kinematic factor
v
that appears in the D -branes amplitudes. While the scattering amplitudes with the
i p
X
usual vertex operators are not gauge invariant, the above amplitudes are invariant.
r From the amplitude describing scattering of two NS-NS states off the O-plane, we
a
find the low energy effective action of O-planes. The result shows a relative factor
2p−6 between couplings to O-planes and to D-branes at (α′)2 order.
1 Introduction
String theory appears to contain various extended objects, other than just strings. D -
p
branes, for instance, are dynamical extended objects that have remarkably simple word
sheet description. World-sheet with boundary on which spacetime coordinate fields satisfy
Dirichlet and Neumann boundary conditions. Using this simple description, it was shown
in [1] that D-branes carry Ramond-Ramond (RR) charges in type II superstring theory.
Another set of extended objects which have simple world sheet description are Orientifold
p-planes. These are non-dynamical planes at the fixed points of spacetime. The world-
sheet in this case has crosscap instead of boundary. Having world sheet description, one
can study these O-planes using well understood tools of two dimensional quantum field
theory. Interaction between O-plane and D-brane, for instance, has been found. From this
amplitude one finds that O-planes have negative tension, and carry RR charge in the case
of Type II superstring theory (see e.g., [2]). We are interested, in this paper, in studying
closed string scattering with O-planes. Closed string scattering off O-planes have been
studied in [3, 4] using boundary state formalism. However, we would like to study them
here using projective plane calculation.
In string theory, the description for scattering of closed string off Dp-brane is as follows:
The string background is taken to be simply flat empty space, however, interactions of
closed stringswitha Dp-branearedescribed by world-sheets withboundary. Theboundary
of world sheet must be fixed to the surface at the position of the Dp-brane. The latter
is accomplished by imposing Neumann boundary conditions on the fields along D -brane
p
world volume, and Dirichlet boundary conditions on the fields transverse to the D -brane
p
[5,2]. The scattering amplitudes describing theinteractions of closed strings with D-branes
has been found in [6, 7, 8, 9], i.e.,
Γ( t/2)Γ( 2s)
AD2 K(1,2) − −
≃ Γ(1 t/2 2s)
− −
where K(1,2) is kinematic factor that depends on momentum and polarization of external
states. It satisfies various Ward identities. In this equation, t is the closed string channel
and s is open string channel.
The description for scattering of closed string off Op-plane is as follows: The string
background is again taken to be flat empty space, however, interactions of closed strings
with these Op-planes are described by world-sheets with a crosscap instead of boundary.
The crosscap must be fixed to the surface at the position of the Op-plane. The latter is
accomplished by choosing appropriate boundary conditions [10]. The scattering amplitude
of closed string off O-plane has been studied in [3, 4] using the boundary states formalism.
In these studies, the standard vertex operator of closed string states, i.e., the same vertex
operatorthatoneusesinscatteringwithD-brane,havebeenused. In[4],ithasbeenargued
that the scattering of NS-NS closed string off O-plane is not gauge invariant because the
standard vertex operators have been used in calculating the amplitude. We note here that
in the presence of O-plane, the vertex operators must satisfy the constraint that when they
1
are on the world volume of O-plane they should describe unoriented states. Using this
constraint, we will find the vertex operators in the presence of O-plane and will find that
the scattering amplitudes describing the interactions of closed strings with O -planes are
p
given by
Γ( t/2)Γ( u/2)
APR2 K(1,2) − − (1)
≃ Γ(1 t/2 u/2)
− −
where K(1,2) is exactly the same kinematic factor that appears in the D-brane case.
Hence, the amplitude is gauge invariant. In this amplitude, both channels are closed string
channel. Here t = (p +p )2 and u = (p +p D)2 where p is momentum of external
1 2 1 2 i
− − ·
state and p D is momentum of its image.
i
·
The paper is organized as follows: In the following section, after writing the NS-NS
vertex operator in the presence of O-plane, we describe the calculation of the scattering
of two such states from an O-plane using conformal field theory techniques. The result
is fully covariant, satisfy the Ward identity and can be written as (1). In section 3, we
repeat the same calculation for scattering amplitude of two RR states from O-plane. In
this case also we show that the result can be written as (1). We then examine the NS-NS
amplitude at low energy in section 4. We show in this section that the massless poles of
the amplitude are in exact agreement with analogous field theory calculations, the contact
terms at (α′)0 order are consistent with negative cosmological constant of O-plane, and
the effective action at (α′)2 order is the same as the effective action of D-brane with the
relative factor 2p−6. We conclude with a discussion of our results in section 5.
2 NS-NS scattering amplitudes
T-duality of unoriented Type I theory gives Type II theory with O-planes at the fixed
points of the T-dual space. In the original Type I theory, states are invariant under world-
sheet parity Ω. In the T-dual theory, this operator has two effects, world sheet parity and
spacetime reflection on the T-dualized coordinates xi. If one considers only the center of
mass dependence of the string wave function, then the projection onto Ω = +1 determines
the wave function at xi to be the same as that at xi, up to a sign1. Various components
−
of massless states in NS-NS sector satisfy [11]
G (xa, xi) = G (xa,xi) ; B (xa, xi) = B (xa,xi)
ab ab ab ab
− − −
G (xa, xi) = G (xa,xi) ; B (xa, xi) = B (xa,xi)
ai ai ai ai
− − −
G (xa, xi) = G (xa,xi) ; B (xa, xi) = B (xa,xi) (2)
ij ij ij ij
− − −
1Our index conventions are that lowercase Greek indices take values in the entire ten-dimentional
spacetime, e.g., µν =0,1,...,9;earlyLatinindices takevaluesinthe worldvolume,e.g., a,b,c=0,1,...,p;
andmiddleLatinindicestakevaluesinthetransversespace,e.g.,i,j =p+1,...,9. Thus,forexample,G
µν
denotes the entire spacetime metric, while G and G denote metric components for directions parallel
ab ij
and orthogonal to the O-planes/D-branes,respectively.
2
Theorientifoldfixedplaneisatxi = 0. Thewavefunctionsthatsatisfytheaboveconditions
are
G = ε eipaxacos(p xi) ; B = iε eipaxa sin(p xi)
ab ab i ab ab i
G = iε eipaxasin(p xi) ; B = ε eipaxacos(p xi)
ai ai i ai ai i
G = ε eipaxacos(p xi) ; B = iε eipaxa sin(p xi) (3)
ij ij i ij ij i
where ε isthepolarizationtensor. Themomentum andpolarizationtensor satisfyp2 = 0,
µν
and pµε = 0 = ε pν. One can rewrite the above wave functions as
µν µν
1
G = ε eip·x +(D ε D) eip·D·x
µν µν µν
2 · ·
1 (cid:16) (cid:17)
B = ε eip·x (D ε D) eip·D·x (4)
µν µν µν
2 − · ·
(cid:16) (cid:17)
where matrix D is η for world volume directions and is η for transverse directions.
ab ij
−
The vertex operator corresponding to the above wave functions is
1
VPR2(ε,p,z,z¯) = ε :Vµ(p,z)::Vν(p,z¯):+(D εT D) :Vµ(p D,z)::Vν(p D,z¯): (5)
2 µν α β · · µν α · β ·
(cid:16) (cid:17)
e e
The holomorphic components Vµ(p,ε) in 0 and -1 picture are given by
α
Vµ (p,z) = e−φ(z)ψµ(z)eip·X(z)
−1
Vµ(p,z) = (∂Xµ(z)+ip ψ(z)ψµ(z)) eip·X(z) . (6)
0 ·
The antiholomorphic components take the same form as in eq. (6) but with the left-moving
fields replaced by their right-moving counterparts – i.e., X(z) X˜(z¯), ψ(z) ψ˜(z¯), and
˜ → →
φ(z) φ(z¯). The vertex operator (5) can also be written as
→
1
VPR2(ε,p,z,z¯) = ε :Vµ(p,z)::Vν(p,z¯): (7)
2 µν α β
(cid:16)
+ε :Vµ(p,z)::eVν(p,z¯):
µν α β ψ(z)↔D·ψ˜(z¯),X(z)↔D·X˜(z¯),φ(z)↔φ˜(z¯)
(cid:16) (cid:17) (cid:19)
e
For the special case that Dµν = ηµν i.e., no T-duality on Type I theory, one should recover
the NS-NS vertex operator of unoriented Type I theory. Interchanging the holomorphic
part with the antiholomorphic part in the second line above, one finds that the two lines
are the same except the fact that the transpose of polarization appears in the second line.
Hence, the Kalb-Ramond antisymmetric tensor is projected out. The surviving states are
graviton and dilaton which are the only NS-NS states in unoriented Type I theory. Note
also that on the O -plane world volume Xi = 0 = ψi. Hence, only matrix Dab = ηab
p
appears in the vertex operator. As a result, as expected, the vertex operator represents
unoriented states on the world volume of O-plane.
3
For D-brane and in the absence of O-planes, there is no constraint on the closed string
states, hence, the wave function for graviton/antisymmetric tensor is
ε eip·x (8)
µν
and the corresponding vertex operator is twice the first term in (5).
The amplitudes describing the scattering of two massless NS-NS states from an O-
plane/D-brane are calculated as two closed string vertex operator insertions on a PR /D
2 2
with appropriate crosscap/boundary conditions [5, 2]. The amplitude is given as
A d2z d2z V (ε ,p ,z ,z¯ ) V (ε ,p ,z ,z¯ ) (9)
1 2 1 1 1 1 1 2 2 2 2 2
≃ h i
Z
both integrals run over the unit disk. The choices for the picture of vertex operators
are arbitrary, however, the sum of superghost charge must be -2. The amplitude can be
separated as
APR2 = A +A +A +A
1 2 3 4
AD2 = A (10)
1
where
1
A ε ε d2z d2z :Vµ (p ,z )::Vν(p ,z¯ )::Vα (p ,z )::Vβ(p ,z¯ ): (11)
1 ≃ 4 1µν 2αβ 1 2h −1 1 1 0 1 1 −1 2 2 0 2 2 i
Z
1
A ε (D εT D) d2z d2z :Vµ (pe ,z )::Vν(p ,z¯ )::Vα (pe D,z )::Vβ(p D,z¯ ):
2 ≃ 4 1µν · 2· αβ 1 2h −1 1 1 0 1 1 −1 2· 2 0 2· 2 i
Z
1
A (D εT D) ε d2z d2z :Vµ (p D,z )e::Vν(p D,z¯ )::Vα (p ,z )e::Vβ(p ,z¯ ):
3 ≃ 4 · 1· µν 2αβ 1 2h −1 1· 1 0 1· 1 −1 2 2 0 2 2 i
Z
1
A (D εT D) (D εT D) e e
4 ≃ 4 · 1· µν · 2· αβ
d2z d2z :Vµ (p D,z )::Vν(p D,z¯ )::Vα (p D,z )::Vβ(p D,z¯ ):
× 1 2h −1 1· 1 0 1· 1 −1 2· 2 0 2· 2 i
Z
e e
To calculate the above correlators, one needs propagator for various world-sheet fields.
Separately, the left- and right-moving fields have standard propagators on sphere world
sheet, e.g.,
Xµ(z)Xν(w) = ηµν log(z w)
h i − −
ηµν
ψµ(z)ψν(w) =
h i −z w
−
φ(z)φ(w) = log(z w) (12)
h i − −
withanalogousexpressionsfortheright-movers(see[12]). However, theprojectiveplane/disk
′
can be obtained as a Z identification of the sphere. This identifies points z and z with
2
′
z = n/z¯ where n = 1 for projective plane and n = 1 for disk. The method of images
− −
4
can then be used to obtain the propagators between left- and right-moving fields [13]. Al-
ternatively, the boundary state of O-plane is obtained by inserting ( 1)m in every terms
−
of the exponent of the D-brane boundary state [10]. This means every z z¯ combination
i j
of D-brane gets replaced by z z¯ [3]. In the propagator between left- and right-moving
i j
−
fields each zw¯ should be replaced by zw¯. Hence,
−
Xµ(z)X˜ν(w¯) = Dµν log(1+nzw¯) (13)
h i −
√nDµν √nDµν
ψµ(z)ψ˜ν(w¯) = ; ψ˜µ(w¯)ψν(z) =
h i −1+nzw¯ h i 1+nzw¯
˜
φ(z)φ(w¯) = log(1+nzw¯) .
h i −
For D -brane case, the fields Xa for a = 0,1, ,p should satisfy Neumann boundary
p
···
conditions, and Xi for i = p + 1,...,9 should satisfy the Dirichlet boundary conditions
[5, 2]. These facts are incorporated by the matrix Dµν. In the scattering amplitude, this
matrix has also the effect of having conservation of momentum along the world volume
of D -brane. To obtain PR propagators, we have replaced zw¯ by zw¯. In the above
p 2
−
propagators, we have fixed the coefficient of the propagator of world sheet fermion by
requiring that the coefficient of unphysical vertex operator in the OPE of Vµ(p ,z ) and
0 1 1
Vν(p ,z¯ ) must be a total derivative [14], i.e.,
0 2 2
e : Vµ(p ,z ) :: Vν(p ,z¯ ) : nηµν p1·D·p2−1 eip1·X+ip2·X˜ +
0 1 1 0 2 2 ≃ (1+nz z¯ )p1·D·p2−2 ···
1 2
e
Given the above propagators, it is now straightforward to evaluate the correlation
functions appearing in the amplitude (11). After evaluating these correlators, one finds
1
A εµνεαβ d2z d2z z z 2p1·p2 1+nz z¯ 2p1·D·p2(1+nz z¯ )p1·D·p1(1+nz z¯ )p2·D·p2
1 ≃ 4 1 2 1 2| 1 − 2| | 1 2| 1 1 2 2
Z
I1 +I2 +I3 +I4 δp+1(p +p D+p +p D) (14)
× µναβ µναβ µναβ µναβ 1 1· 2 2·
(cid:16) (cid:17)
where delta function gives conservation of momentum along world volume of O-plane/D-
brane. There is a factor of i(2π)p+1CX Cψ Cφ for PR case and i(2π)p+1CX Cψ Cφ
PR2 PR2 PR2 2 D2 D2 D2
for disk where CX , for example, is the functional determinant for the string field Xa [13].
PR2
We are not trying to calculate these constants here. Hence, we omit them here and at the
end normalize the amplitude by comparing with low energy effective action. In the above
equation,
η η p D nz p D nz p
I1 = µα νβ + 1· ν 1 2· ν 2 + 2ν
µναβ (z1 z2)2 (z¯1 z¯2)2 "−(1+nz1z¯1) − (1+nz2z¯1) (z¯2 z¯1)#
− − −
p D nz p D nz p
1 β 1 2 β 2 1β
· · +
×"−(1+nz1z¯2) − (1+nz2z¯2) (z¯1 z¯2)#!
−
n p D nz p D nz p
I2 = 1· β 1 2· β 2 + 1β
µναβ (z1 z2) "−(1+nz1z¯2) − (1+nz2z¯2) (z¯1 z¯2)#
− −
5
p D D p D D
1 µ να 1 α µν
· − ·
× (1+nz1z¯1)(1+nz2z¯1)!
n p D nz p D nz p
I3 = 1· ν 1 2· ν 2 + 2ν
µναβ (z1 z2) "−(1+nz1z¯1) − (1+nz2z¯1) (z¯2 z¯1)#
− −
p D D p D D
2 µ αβ 2 α µβ
· − ·
× (1+nz1z¯2)(1+nz2z¯2) !
n p D (η p D p D ) D (p D p p p D )
I4 = 1· µ νβ 2 · α − 2ν αβ − µν 2 · α 1β − 1 · 2 αβ
µναβ (z1 z2) (1+nz1z¯1)(z¯1 z¯2)(1+nz2z¯2)
− −
p D ( η p D +p D )+D (p D p p p D )
2 µ νβ 1 α 1β αν µβ 1 α 2ν 1 2 αν
+ · − · · − ·
(1+nz z¯ )(z¯ z¯ )(1+nz z¯ )
1 2 1 2 2 1
−
η (p p η p p )
µα 1 2 νβ 1β 2ν
n · − (15)
− (z1 z2)(z¯1 z¯2)2 !
− −
Now the integrand should be invariant under the CKG of the projective plane/disk
which is SL(2,C) group of sphere that respects the identification z = n/z¯. Transforma-
−
tion of z and z¯ under this group are
az +b dz¯ nc
z ; z¯ n − (16)
→ cz +d → − bz¯ na
−
where a,b,c,d are four complex parameters that satisfy ad cb = 1. Moreover, comparing
−
the second transformation above with the complex conjugate of first transformation gives
¯ ¯
b = nc and d = a. Under above transformation, one finds
−
z z
1 2
z z −
1 2
− → (cz +d)(cz +d)
1 2
z¯ z¯
1 2
z¯ z¯ −
1 2
− → (bz¯ na)(bz¯ na)
1 2
− −
1+nz z¯
1 2
1+nz z¯
1 2
→ (cz +d)(bz¯ na)
1 2
−
dzdz¯
dzdz¯ (17)
→ (cz +d)2(bz¯ na)2
−
One can check that the integrand in the amplitude (14) is invariant under the above
transformations. In checking this, one should replace one of the momentum in each bracket
[...] in (15) in terms of other two momentum using conservation of momentum and the
physical condition on the polarization of external states, i.e., p ε = 0. To gauge fix this
i i
·
symmetry, we first fix one vertex operator at z = z¯ = 0. The residual symmetry is then
1 1
z az which is a rotation in the z-plane. To fix the residual symmetry, we fix the polar
→ a¯
coordinate of the second vertex operator, i.e., z = z¯ = r. Ignoring the volume of the
2 2
| | | |
symmetry group, i.e.,
d2z d2z dr2 (18)
1 2
→
6
one is left with a single real integral of the form
iκT 1 a a a
AD2 = p dr2 (r2)p1·p2(1 r2)p2·D·p2 1 + 2 + 3
2 − r2 1 r2 r4
′ Z0 (cid:20) − (cid:21)
iκT 1 a a a
APR2 = p dr2 (r2)p1·p2(1+r2)p2·D·p2 1 + 2 + 3 (19)
1 8 r2 1+r2 r4
Z0 (cid:20) (cid:21)
where a , a and a are three kinematic factors depending only on the spacetime momenta
1 2 3
and polarization tensors. We have also normalized the amplitudes at this point by the
′
introduction of factors of κ, the closed string coupling, and T (T ), the O-plane (D-brane)
p p
tension, respectively.2 Note that the constants that we have omitted in (14) and in (18)
′
have the same sign for both PR and D . Hence, the constants T and T must have the
2 2 p p
same sign. The kinematic factors are
a = Tr(ε εT)(1+t/2)
3 1· 2
a = n Tr(ε D)p ε p p ε D ε p p ε εT D p
1 1· 1· 2· 1 − 1· 2· · 1· 2− 1· 2· 1· · 1
(cid:16) 1
p εT ε D p (p εT ε p +p ε εT p )
− 1· 2· 1· · 1− 2 2· 1· 2· 1 1· 2· 1· 2
t
(Tr(ε D)Tr(ε D) Tr(ε D ε D))+ 1 2
1 2 1 2
−4 · · − · · · ←→
n o(cid:19)
a = Tr(ε D)(p ε D p p D ε D p )
2 1 1 2 2 2 2 1
· · · · − · · · ·
+p D ε D ε D p p D ε εT D p
1· · 1· · 2· · 2− 2· · 2· 1· · 1
+Tr(ε D)Tr(ε D)(t/4)+ 1 2 (20)
1 2
· · ←→
n o
where t = (p +p )2 = 2p p is the momentum transfer to the O-plane/D-brane, and
1 2 1 2
− − ·
s = papbη = 1p D p is the momentum flowing parallel to the world-volume of the
− 1 1 ab −2 1· · 1
O-plane/D-brane. Our notation is such that e.g., p ε εT D p = pµε ε λν D pρ.
1· 2· 1· · 1 1 2µν 1 λρ 1
Using the following integrals:
1 F(β,1 α;2 α; 1)
dxx−α(1+x)−β = − − −
1 α
Z0 −
1
dxx−α(1 x)−β = B(1 α,1 β) (21)
− − −
Z0
One finds
iκT
AD2 = p (a B( 1 t/2,1 2s) (22)
3
2 − − −
+a B( t/2,1 2s)+a B(1 t/2, 2s)
1 2
− − − −
(cid:19)
2Here and in the subsequent amplitudes, we omit the Dirac delta-function which imposes momentum
conservationin the world-volumedirections. We haveintroducedaphase i thoughwhichcorrespondsto
−
that of the analogous field theory amplitudes calculated in Minkowski space — see sect.4.
7
′
iκT F(2s, 1 t/2; t/2; 1)
APR2 = p a − − − − (23)
1 8 3 1 t/2
− −
F(2s, t/2;1 t/2; 1) F(2s+1,1 t/2;2 t/2; 1)
+a − − − +a − − −
1 2
t/2 1 t/2 !
− −
As is evident the final amplitude is symmetric under the interchange of the two string
states, i.e., 1 2, despite the asymmetric appearance of the initial integrand in eq. (9).
←→
From the gamma function factors appearing D amplitude, we see that the amplitudes
2
contain two infinite series poles3 corresponding to closed string states in the t-channel with
α′m2 = 4n, and to open string states in the s-channel with α′m2 = n, with n = 0,1,2,....
However, the PR amplitude has poles only in the t-channel with α′m2 = 4n. Having
2
no open string pole is consistent with the fact that the O -plans, unlike the D -branes,
p p
are not dynamical objects. The amplitude AD2 is the one that has been found in [9] using
upper-halfz-planecalculation. WhilethisamplitudesatisfiestheWardidentitiesassociated
with the gauge invariance of external states, i.e., the amplitude vanishes upon substituting
ε p q or q p , where q p = 0 [9], the amplitude APR2 does not satisfy the Ward
iµν → iµ iν iµ iν i· i 1
identities. Using the package [15] for expanding the hypergeometric functions, one finds
that APR2 has the following expansion at low energy4:
1
′
iκT 1
APR2 = p (t 4s)Tr(ε εT)+2a
1 8 −t − 1· 2 1
(cid:18) h i
+ln(2) 4sTr(ε εT)+a
1· 2 2
π2 h i
+ 2s(t+4s)Tr(ε εT) 4sa +ta (ln(2))2 4s2Tr(ε εT)+sa
24 1· 2 − 1 2 − 1· 2 2
h i h i
+ (24)
···
(cid:19)
The massless pole is reproduced in field theory assuming linear coupling of gravity to O-
plane. However, the (α′)0 contact term is not consistent with the cosmological constant
of the O-plane effective action. Moreover, the α′ and (α′)2 contact terms above are not
consistent with what is expected for the effective action of O-plane. This part of amplitude
has been also found in [4] using the boundary state formalism. However, they have used
another picture for the graviton vertex operator, i.e., they have used V whereas we use
(0,0)
V(−1,0). Their result is the same as AP1R2 above up to some contact terms. It has been
argued in [4] that the ln(2) terms should be absent in the final gauge invariant amplitude
since their presence would be in conflict with the fact that such terms are not present in
the effective actionof type I string theory. Using this, anexpansion for the amplitude APR2
1
has been found in [4] that is consistent with expected low energy effective action. We will
show shortly, however, that when we add the contributions from A , A and A the final
2 3 4
result satisfies the Ward identities, it does not contain the ln(2) terms, it is independent of
3We explicitly restore α′ here. Otherwise our conventions set α′ =2.
4I would like to thank H. Ghorbani for finding this expansion.
8
the choice for the picture of vertex operators, and it produces expected low energy effective
action.
Before calculating A , A , A , we would like to compare the t-channel poles in the
2 3 4
amplitudes (22) and (23). The Feynman rule for calculating the t-channel poles in field
theory is as follows: It is a summation of infinite terms. Each is a product of a vertex
of three states where two of them are the on-shell external states, the propagatore of the
off-shell state, and the coupling of the off-shell state to O-plane/D-brane. Using the fact
that the external states are the same in both amplitudes, the t-channel analysis gives
information about the linear coupling of off-shell closed string states to O-plane and to
D-brane. To compare them, consider the following definition of the beta function (see e.g.,
[16]):
∞ ( 1)n(β 1)(β 2) (β n)
B(α,β) = − − − ··· − (25)
n! α+n
n=0
X
This expansion can be rewritten as
∞
1 Γ(n+1 β)Γ(n+α)
B(α,β) = −
Γ(1 β) Γ(n+α+1)
− nX=0
1
= F(1 β,α;1+α,1) (26)
α −
From this, one finds the following expansion for the hypergeometric function appearing in
(23):
∞
1 1 (β 1)(β 2) (β n)
F(1 β,α;1+α, 1) = − − ··· − (27)
α − − n! α+n
n=0
X
Using (25) and (27), one finds the t-channel expansion for the amplitudes in (22) and (23).
Noting that aD2 = aPR2, one finds that the two amplitudes have the same pole structure,
1 − 1
however, the overall sign of each pole in two amplitudes is different . The signs are such
that the massless poles have opposite sign, the first massive poles have the same sign, the
second massive poles have again opposite sign, and so on. This observation can also be
made from the interaction of D-brane with O-plane [2].
NowtocalculateA , A , A , onemaytrytofindthemfromA byreplacing appropriate
2 3 4 1
momenta and polarization tensors, e.g., if (p ,ε ,p ,ε ) (p ,ε ,p D,D εT D), then
1 1 2 2 → 1 1 2· · 2 ·
A A . In this way, using the identity (D D)µν = ηµν, one can easily show that
1 2
→ ·
A = A ; A = A (28)
1 4 2 3
Finding A in this way, the final result, APR2, would be in terms of some hypergeometric
2
functions which we have found it difficult to simplify the final result. Instead, we calculate
A using another arrangement for the picture of the vertex operators in (11). We write it
2
as
A ε (D ε D) d2z d2z :Vµ (p ,z )::Vν(p ,z¯ )::Vβ(p D,z )::Vα (p D,z¯ ):
2 ≃ 1µν · 2· αβ 1 2h −1 1 1 0 1 1 0 2· 2 −1 2· 2 i
Z
e e
9